R9, N12NumberFoundation & HigherGrades 2–5

Percentages

"Per cent" means "out of 100", so 17% is $\frac{17}{100}$ or $0.17$ . Percentages are the most common real-world maths on GCSE papers: discounts, pay rises, interest, test scores and survey results. The key tool is the decimal multiplier, which turns every percentage question into a single multiplication.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Without a calculator, build percentages from easy pieces: 10% is dividing by 10, 5% is half of that, 1% is dividing by 100. With a calculator, use the multiplier: 17% of £240 is $0.17 \times 240$ .

Increasing by 15% leaves you with 115% of the original, so multiply by $1.15$ . Decreasing by 15% leaves 85%, so multiply by $0.85$ . That one step replaces "find the percentage, then add or subtract", and it is essential for compound interest later.

Finding a Percentage of an AmountCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Find a percentage of an amount with and without a calculator
Write one quantity as a percentage of another
Increase and decrease by a percentage using multipliers
Calculate simple interest and compare offers

Percentage of an amount

Non-calculator: find 10% (divide by 10), 5% (half of 10%) and 1% (divide by 100), then add pieces. For 35%: $10% \times 3 + 5%$ .

Calculator: convert the percentage to a decimal and multiply. $12.5%$ of 64 is $0.125 \times 64 = 8$ .

110% = £26, so 30% = £78.

25% = £13.

335% = £78 + £13.

Answer£91

Your turnSame method, new numbers — try it before moving on.

Without a calculator, find 15% of £340.

Your answer

Your turnSame method, new numbers — try it before moving on.

Find 8% of 2,350 kg.

Your answer

Tip — Money answers need 2 decimal places: £14.70, not £14.7.

One quantity as a percentage of another

Write the part as a fraction of the whole, then multiply by 100. Make sure both are in the same units first.

\frac{42}{56} \times 100

= 0.75 \times 100 = 75%

Answer

75%

Your turnSame method, new numbers — try it before moving on.

What is 45 cm as a percentage of 3 m?

Your answer

Percentage increase and decrease

The multiplier for an increase of $p %$ is $1 + \frac{p}{100}$ ; for a decrease it is $1 - \frac{p}{100}$ .

Increase 80 by 12%: $80 \times 1.12 = 89.6$ . Decrease 80 by 12%: $80 \times 0.88 = 70.4$ .

An increase of 15% leaves 115% of the original; a decrease of 15% leaves 85%.

1Multiplier:

1 - 0.22 = 0.78

450 \times 0.78 = 351

Answer£351

Your turnSame method, new numbers — try it before moving on.

A salary of £28,500 increases by 3.5%. What is the new salary, in pounds?

Your answer

Why the order of changes matters

Increasing by 20% and then decreasing by 20% does not get you back to the start: $100 \times 1.2 \times 0.8 = 96$ . The decrease is 20% of a bigger number. Multipliers make this obvious.

Simple interest and best buys

Simple interest pays the same amount every year: interest = principal × rate × years. £2000 at 3% for 4 years earns $2000 \times 0.03 \times 4 = £240$ .

To compare offers, work out the final price or value for each and state which is better with the figures.

1A:

80 \times 0.75 = £60

2B:

\frac{1}{5}

of 75 is 15, so £60.

AnswerThey cost the same — £60.

Your turnSame method, new numbers — try it before moving on.

£1,800 is invested at 2.5% simple interest per year. How much interest is earned in 6 years, in pounds?

Your answer

Think like an examiner

In "which is the better buy" questions, the final mark needs a conclusion supported by figures for both options.
Write the multiplier you use ( $0.78$ , $1.035$ ) — it earns the method mark even if you mistype into the calculator.
Give money to 2 decimal places and include units.

Percentages

p % of A = \frac{p}{100} \times A

Percentage of an amount.

\frac{part}{whole} \times 100

One quantity as a percentage of another.

A \times (1 \pm \frac{p}{100})

Increase or decrease by

p %

Watch out for these

Using

1.5

as the multiplier for a 5% increase.

5% increase is

1.05

Decreasing by 22% using

0.22

That finds the discount. The sale price uses

0.78

45 cm as a percentage of 3 m:

\frac{45}{3} \times 100

Same units:

\frac{45}{300} \times 100 = 15%

Thinking +20% then −20% returns to the original.

1.2 \times 0.8 = 0.96

— a 4% overall loss.

Stretch yourself

In a sale, prices are reduced by 30%. On the last day, the sale prices are reduced by a further 30%. Is this the same as a 60% reduction? If not, what is the overall percentage reduction?

Hint — Apply both multipliers to a starting price of £100.

Questions students ask

How do I find 1% without a calculator?

Divide by 100. Then any whole percentage is a multiple of 1%, e.g. 7% is

7 \times 1%

Can a percentage be more than 100?

Yes. A price rising from £20 to £50 is 250% of the original. You cannot, however, take away more than 100% of something.

What is the difference between simple and compound interest?

Simple interest is calculated on the original amount each year. Compound interest is calculated on the growing total — see Percentage Change & Growth.

Key takeaways

Non-calculator: build from 10%, 5% and 1%.
Calculator: multiply by the decimal equivalent.
Part as a percentage: $\frac{part}{whole} \times 100$ , same units.
Increase by $p %$ : × $(1 + \frac{p}{100})$ ; decrease: × $(1 - \frac{p}{100})$ .
Simple interest = principal × rate × years.

Exam-style questions

Practise it the way it is examined

3 original questions · 11 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Work out 45% of 620 without a calculator.[2]

Question 2Grade 45 marks

Priya earns £2,150 a month. She spends £731 on rent.

(a)What percentage of her monthly pay does she spend on rent?[2]
(b)Her pay rises by 4%. Her rent rises by 6%. What percentage of her new pay is her new rent? Give your answer to 1 decimal place.[3]

Question 3Grade 54 marks

Two phone contracts run for 24 months. Contract A: £180 up front, then £26 per month. Contract B: no up-front cost, £38 per month, with 15% off the total.

Which contract is cheaper overall, and by how much?[4]

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Percentages

Percentage of an amount

One quantity as a percentage of another

Percentage increase and decrease

Simple interest and best buys

Practise it the way it is examined

Skill drill

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Related topics

Percentages

Percentage of an amount

One quantity as a percentage of another

Percentage increase and decrease

Simple interest and best buys

Practise it the way it is examined

Skill drill

Practise this topic

Related topics